Output Feedback Control for Spacecraft with Coupled Translation and Attitude Dynamics Haizhou Pan, Hong Wong, and Vikram Kapila Mechanical, Aerospace, and Manufacturing Engineering, Polytechnic University, Brooklyn, NY 11201 Abstract— In this paper, we address a tracking control problem for a spacecraft with coupled translation and attitude motion, in the absence of translation and angular velocity measurements. We begin by describing the mutually coupled translation and attitude dynamics of the spacecraft. Next, a suitable high-pass filter is employed to estimate the spacecraft translation and angular velocities using measurements of its translational position and attitude orientation. Using a Lyapunov framework, a nonlinear output feedback control law is designed that ensures the semi-global asymptotic convergence of the spacecraft translation and attitude position tracking errors, despite the lack of translation and angular velocity feedback.
I. Introduction Recent years have witnessed a growing interest in future space missions such as capture and removal of orbital debris [9], [11] and orbital rendezvous with maneuvering target. Such space missions rely on highly maneuverable spacecraft necessitating the development of a systematic framework for simultaneous control of translation and attitude motion of the spacecraft [8]. In contrast to the study of six-DOF rigid body dynamics and control in the realm of aircraft and underwater vehicles [2], [10], the six-DOF rigid body dynamics and control problem for spacecraft has received scant attention. Some recent exceptions include [8], [9], [11]. A typical feature of the aforementioned control designs is the requirement that the translation and angular velocity measurements of spacecraft be available for feedback. Unfortunately, this requirement is not always satisfied in practice since, e.g., a low sensor count may be desirable to keep cost/weight low. In prior research, several authors have addressed the problem of output feedback control of spacecraft attitude. Specifically, using a three-parameter representation of the spacecraft attitude, a passivity-based controller was developed in [7]. Furthermore, using the modified Rodrigues parameter representation of the spacecraft attitude, an adaptive output feedback attitude tracking controller was developed in [12]. Finally, using the four-parameter quaternion representation of the spacecraft attitude, an adaptive output feedback attitude tracking controller was developed in [4]. However, the output feedback control problem for the six-DOF coupled translation and attitude motion of spacecraft remains to be addressed. In this paper, we address the output feedback tracking control problem for the six-DOF motion of a spacecraft using the coupled translation and attitude dynamics of a spacecraft developed in [8]. A high-pass filter is employed to generate a velocity-related signal from the translational position and attitude orientation Research supported in part by the National Aeronautics and Space Administration–Goddard Space Flight Center under Grant NGT5-151 and the NASA/New York Space Grant Consortium under Grant 323105891.
measurements. A judicious modification of the generally recommended [5] filter is implemented to overcome the complexity arising from the mutual coupling of spacecraft translation and attitude dynamics. Using a suitable Lyaponuv function, our nonlinear output feedback control law guarantees asymptotic convergence of the translation and attitude position tracking errors, despite the lack of translation and angular velocity feedback. II. Mathematical Preliminaries Throughout this paper, several reference frames are employed to characterize the translation and attitude dynamics of a spacecraft. Each reference frame used in this paper is assumed to consist of three basis vectors which are right-handed, mutually perpendicular, and of unit length. Let F denote a reference frame and → → → let i , j , and k denote the three basis vectors of F . →
Then F
=
→ i → j → k
denotes the vectrix of the reference →
frame F [6]. A vector A can be expressed in the → → → → reference frame F as A = a1 i +a2 j +a3 k , where → → a1 , a2 , and a3 denote the components of A along i , → → j , and k , respectively. Frequently, we will assemble T these components as A = [a1 a2 a3 ] . Using the above vectrix formalism and the usual vector inner product, → a vector A can be expressed in the reference frame F → → → as A = AT F =F→T A. The vectrix F has the following two properties: i) → → F · F T = I3 , where “·” denotes the usual vector dot product and In denotes an →n dimensional iden→ → →
→
tity matrix and ii) F × F T =
0 → − k → j
k → 0 → − i
− j → i → 0
, where
“×” denotes the usual vector cross product. Thus, it → → → → → follows that A =F · A=A · F . Next, let B be a → → → vector, which is expressed in F by B = b1 i +b2 j → T +b3 k , and let B = [b1 b2 b3 ] . Then, it follows that → → → → → A · B = AT B = B TA and A × B =F T A× B, where 0 −a3 a2 a3 0 −a1 A× = . −a2 a1 0 Throughout this paper, various vectors will be expressed in two or more different reference frames using the rotation matrix concept. For example, consider → a vector A expressed in the reference frame Fu has → → → components Au , i.e., A=F uT Au , where F u denotes the vectrix of the reference frame Fu . Similarly, consider → the vector A expressed in the reference frame Fv has
→
→
→
components Av , i.e., A =F vT Av , where F v denotes the vectrix of the reference frame Fv . Then, it follows that → → → → T Av =F v · F uT Au = Cvu Au , where Cvu = Fv · F u ∈ SO(3) denotes the rotation matrix that transforms the components of a vector expressed in Fu (viz., Au ) to the components of the same vector expressed in Fv (viz., Av ). Here the notation SO(3) represents the set of all 3 × 3 rotation matrices. → Let us consider the vector A expressed in another → → T reference frame Fw has components Aw , i.e., A=F w Aw , → where F w denotes the vectrix of the reference frame → → T Fw . Then, as above, Au =F u · F w Aw = Cuw Aw → → → T and Av =F v · F w Aw = Cvw Aw , where Cuw = Fu → → → T T · Fw ∈ SO(3) and Cvw = F v · F w ∈ SO(3). Finally, it u w follows that Av = Cv Cu Aw and Cvw = Cvu Cuw . III. Spacecraft Dynamic Modeling In this section, we review a nonlinear model characterizing the translation and attitude dynamics of a spacecraft [8]. The spacecraft is modeled as a rigid body with actuators that provide body-fixed forces and torques about three mutually perpendicular axes that define a body-fixed reference frame Fb located at the mass center of the spacecraft as shown in Figure 1. In this paper, we fully account for the mutual coupling between the translation and attitude motion of the spacecraft. For a given desired translation and attitude motion trajectory, we develop the translation and attitude error dynamics of the spacecraft. Finally, we state our control objective for the translation and attitude motion of the spacecraft. A. Spacecraft Translation Motion Dynamics Let Fi be an inertial reference frame fixed at the → center of the earth and let A be an arbitrary vector measured with respect to the origin of Fi . Then, in •
→
→
this paper, A (t) denotes the time derivative of A (t) measured in Fi . Using this notation, the translation motion dynamics of the spacecraft is given by [6] •
→
→
R = V,
•
→
→
→
•
→ Rd
→
m V = fe + f d − f ,
(1)
where m denotes the mass of the spacecraft, R (t) and → V (t) denote the position and velocity of its mass center, →
fe (t) denotes the inverse-square gravitational force that →
leads to an elliptical orbit [3], [6], fd (t) denotes the attitude-dependent disturbance force that causes the spacecraft trajectory to deviate from an ellipse [6], and →
f (t) denotes the external control force. The inversesquare gravitational force and the attitude-dependent → → disturbance force are characterized as fe = − µm R and → 3 || R → → → ||→ → → → → fd = − 3µ tr (J) I +2 J · Z −5 Z · J · Z Z , re→ spectively, where µ = M G with M being the mass of the earth and G being the universal gravitational constant, J is the constant, positive-definite, symmetric → inertia matrix of the spacecraft expressed in Fb , J → → T = F b J F b denotes the central inertia dyadic of the
•
→
→
µ
→
= Vd,
Vd +
||
→ Rd
||3
→ Rd =
0,
(2)
→
where Rd (t) and V d (t) denote the desired position and velocity of the spacecraft mass center. → Next, let ω b (t) denote the angular velocity of Fb ◦
→
relative to Fi . In this paper, A (t) denotes the time → derivative of an arbitrary vector A measured in Fb . → Using this notation, the time derivative of vector A measured in Fi is given by •
◦
→
→
→
→
(3) A = A + ωb × A . Now we develop the translation motion error dynamics of the spacecraft. Before proceeding, for convenience, we introduce the notation → eR
=
→ Rd
→
→ → eV = Vd
− R,
→
−V .
(4)
Computing the time derivative on both sides of the two equations in (4) measured in Fi and using (3), we obtain ◦
◦
• → → → e R = Rd − R
→
→
•
→
◦
→
◦
→
→
→
+ ω b × eR , eV =V d − V + ω b × eV . (5) ◦
→
◦
→
◦
→
Note that from the first equation of (4), eR = Rd − R •
•
•
→
→
→
and eR = Rd − R. Similarly, from the second equation ◦
→
◦
◦
→
→
→
→
•
→
•
→
•
→
of (4), eV = V d − V and eV = V d − V . Combining (1), (2), (4), and (5), we obtain the translation motion error dynamics of the spacecraft given by ◦
→ → eR = eV
− ω b × eR ,
→ µR → → → eV =− ω b × eV − → d || Rd ||3 ◦
→
2|| R ||4
→
spacecraft [6], I denotes the dyadic of a 3 × 3 identity → → → → → R matrix, Z with R = R · R , and tr (·) = → || R || denotes the trace of a matrix. In this paper, the desired translation motion dynamics of the spacecraft is assumed to track an elliptical orbit given by
(6)
→ → → 1 1 − f + fd + f. (7) m e m
Now using the framework of Section II, various vectors of interest can be expressed in the spacecraft body-fixed reference frame Fb as follows → R R˙ Rd R˙ d V Vd eR e˙ R eV e˙ V ωb f fe fd = Fb ◦ ◦ ◦ ◦ → → → → → → → → → → → → → → · R R Rd Rd V V d eR eR eV eV ω b f fe fd , (8)
˙ where R(t), R(t), Rd (t), R˙ d (t), V (t), Vd (t), eR (t), e˙ R (t), → eV (t), e˙ V (t), ωb (t), f (t), fe (t), fd (t) ∈ R3 and F b denotes the vectrix of the reference frame Fb . Next, an application of the vectrix formalism of Section II on (6) and (7) yields e˙ R = eV − ωb× eR ,
µ µ 3µ Rd + R+ e˙ V = −ωb× eV − 3 3 ||Rd || ||R|| 2m||R||4
(9)
R 1 5RT JR + f . (10) · tr (J) I3 + 2J − I 3 ||R||2 ||R|| m Remark 3.1: As in [8], the control design framework of Section IV can handle arbitrary time-varying desired → → trajectories Rd (t) under the assumption that Rd (t) and its first two time derivatives are all bounded functions of time. B. Spacecraft Attitude Dynamics The attitude dynamics of the spacecraft is given by •
→
→
→
h = τg + τ ,
→
→
→
(11)
→ ωb
where h (t) given by h = J · denotes the angular momentum of the spacecraft about its mass center, → → → → 3µ → τg (t) given by τg = → Z × J · Z denotes the gravity || R ||3 →
gradient torque [6], and τ (t) denotes the control torque. •
→
◦
→
Using (3) we obtain h = h + using the framework of Section interest can be expressed in the reference frame Fb as → → h ω˙ b τg τ F b · h =
→
→ ωb
× h . Once again, II, various vectors of spacecraft body-fixed ◦
→ → → ωb τg τ
,
(12)
where h(t), ω˙ b (t), τg (t), τ (t) ∈ R3 . Next, an application of the vectrix formalism of Section II on (11) yields J ω˙ b = −ωb× Jωb + τg + τ.
(13)
Now we use the nonsingular, four-parameter quaternion representation to relate the time derivative of the spacecraft angular orientation to the angular velocity ωb as follows [6] T ε˙Tb ζ˙bT = E(εb , ζb )ωb , (14) × ε + ζb I3 1 b where E(εb , ζb ) and [εb (t), ζb (t)] ∈ R3 T =2 −εb
×R represents the quaternion, which characterizes the attitude of Fb with respect to Fi . By construction, the quaternion [εb , ζb ] must satisfy the unit norm constraint εTb εb + ζb2 = 1. Following [6], the rotation matrix Cbi ∈ SO(3) that brings the inertial frame Fi onto the spacecraft body-fixed b is given 2 reference frame F × T T as Cbi = C(εb , ζb ) = ζb − εb εb I3 + 2εb εb − 2ζb εb . The dynamic and kinematic equations of (13) and (14) represent the attitude dynamics of the spacecraft. Next, we characterize the desired attitude of the spacecraft using a desired, spacecraft body-fixed ref→ erence frame Fd . Let ωd (t) denote the desired angular
→ ωd (t)
denote velocity of Fd with respect to Fi and let → the time derivative of ωd measured in Fd . Using the →
→
framework of Section II, we express ωd and ωd in the desired, spacecraft body-fixed reference frame Fd as follows → → → ωd , ωd (t), ω˙ d (t) ∈ R3 , (15) ωd ω˙ d = F d · ωd →
where F d denotes the vectrix of the reference frame Fd . The angular orientation of the desired, spacecraft body-
fixed reference frame Fd with respect to the inertial frame Fi is characterized by the desired unit quaternion [εd (t), ζd (t)] ∈ R3 × R, whose kinematics are governed by T ε˙Td ζ˙dT = E(εd , ζd )ωd . (16) The rotation matrix Cdi ∈ SO(3) that brings the reference frame Fi onto the reference frame Fd is given by Cdi = C(εd , ζd ). In addition, using (16), it ˙d = follows that ωd = 2(ζd ε˙d − ζ˙d εd ) − 2ε× d ε˙d and ω 2(ζd ε¨d − ζ¨d εd )−2ε× ε ¨ . In this paper, we assume that εd , d d ζd , and their first two time derivatives are all bounded functions of time, which yields boundedness of ωd and ω˙ d given above. Now we develop the attitude error dynamics of the spacecraft. Let [eε (t), eζ (t)] ∈ R3 × R denote the quaternion characterizing the mismatch between the actual orientation of the spacecraft Fb and desired orientation of the spacecraft Fd . By construction, the error quaternion [eε , eζ ] must satisfy the unit norm constraint eTε eε + e2ζ = 1. In addition, [eε , eζ ] can be characterized using [εb , ζb ] and [εd , ζd ] as follows T eε = ζd εb − ζb εd + ε× b εd and eζ = ζd ζb + εd εb [1], d [4], [6]. The corresponding rotation matrix Cb ∈ SO(3) that brings the desired, spacecraft body-fixed reference frame Fd onto the spacecraft body-fixed frame Fb is given by T
Cbd = C(eε , eζ ) = Cbi Cdi .
(17)
→
Next, let ω (t) denote the angular velocity of Fb with respect to Fd . Then, it follows that →
→
→
ω = ω b − ωd .
(18) →
Now using the framework of Section II, we express ω in the spacecraft body-fixed reference frame Fb as ω → ωb = F b
= → ωb
→
→
F b · ω,
ω(t) ∈ R3 .
· from (8), Using and (19), Eq. (18) yields
→ ωd = F d
·
(19) → ωd
from (15),
ω = ωb − Cbd ωd . Note that by applying (3) to •
◦
→ → ωb = ωb
•
→
◦
→ ωb
(20) →
and ω we get
→
→ → and ω = ω + ω b × ω , respectively. Similarly, • → → it can be shown that ωd = ωd . Now computing the time derivative of (18) measured in Fi and performing simple
manipulations, it can be shown that the time derivative ◦
→
→
◦
→
→
→
→
of ω measured in Fb is given by ω = ωb − ω b × ω − ωd . ◦
→
Expressing ω in the spacecraft body-fixed reference frame Fb as ω˙
=
→ Fb
◦
→
˙ ∈ R3 , and using · ω , ω(t) ◦
→
(12), (15), (19), and (20), we can now express ω in the spacecraft body-fixed reference frame Fb . Finally, multiplying the resultant expression by J on both sides, we obtain J ω˙ = J ω˙ b + Jω × Cbd ωd − JCbd ω˙ d . (21) We now use (13), (20), and (21) to obtain the following open-loop attitude tracking error dynamics
of the spacecraft × J ω˙ = − ω + Cbd ωd J ω + Cbd ωd 3µ +J ω × Cbd ωd − Cbd ω˙ d + R× JR + τ. (22) ||R||5 In addition, the open-loop attitude tracking error kinematics is given by [1], [4], [6] T T = E(eε , eζ )ω. (23) e˙ ε e˙ Tζ
we introduce two matrices T (t), P (t) ∈ R3×3 defined as eζ −eε3 eε2 1 −1 eε eζ −eε1 , P T , (29) = T = 2 −e3ε eε eζ
Note that the attitude dynamics of (22) can be rearranged to yield
Next, we define the position and velocity tracking error variables r0 (t), v0 (t) ∈ R6 as follows T T T T T T , v0 . (31) r0 = eR eε = eV e˙ ε
J ω˙ = −ω × Jω + τun + τkn + τ,
(24)
where τun (t), τkn (t) ∈ R are defined as d × d × d × τun ω, (25) = (JCb ωd ) − (Cb ωd ) J − J(Cb ωd ) 3µ d × d d τkn ˙ d+ R× JR. (26) = −(Cb ωd ) J(Cb ωd )−JCb ω R5 3
Remark 3.2: The definition of τun in (25) depends on ω, which is not measured. Thus, τun can not be used in the control design. On the other hand, the definition of τkn in (26) depends on the desired attitude trajectory and the translational position R, signals that are known/measured. Thus, τkn can be used in the control design. C. Control Objectives In this paper, the control objective for the translation motion dynamics of the spacecraft requires that the mass center of the spacecraft track the desired → → translation motion trajectory, i.e., R (t) →Rd (t) as •
•
→
→
t → ∞. In addition, it is required that R (t) →Rd (t) as t → ∞. Using (1), (2), (4), and (8), the spacecraft translation motion tracking control objective can be stated as follows lim eR (t), eV (t) = 0. (27) t→∞
The control objective for the attitude dynamics of the spacecraft requires that the actual attitude of the spacecraft track the desired attitude trajectory, i.e., the rotation matrix Cbi must coincide with the rotation matrix Cdi in steady-state. Using (17), this control objective can be equivalently characterized as lim Cbd = →
t→∞ →
I3 . Furthermore, it is required that ω b (t) → ωd (t) as t → ∞. With the aid of the unit norm constraint of (eε , eζ ) and using (17)–(19), the spacecraft attitude tracking control objective can be equivalently stated as follows lim eε (t), ω(t) = 0.
t→∞
(28)
The control objectives of (27) and (28) are to be met under the constraint of no direct velocity feedback (i.e., V and ωb are not measured). IV. Spacecraft Output Feedback Control Design In this section, we develop an output feedback controller based on the system dynamics of (9), (10), (23), and (24) such that the tracking error variables eR , eV , eε , and ω exhibit asymptotic stability. Before we proceed with the control design, for notational convenience,
2
1
where eε1 (t), eε2 (t), eε3 (t) ∈ R are the components of eε . Using (29), e˙ ε of (23) can be written in a compact form as follows e˙ ε = T ω, ⇒ P e˙ ε = ω. (30)
Now differentiating r0 in (31) with respect to time and using (9) and (31), we obtain T r˙0 = e˙ TR e˙ Tε = v0 − Ω, (32) T × where Ω(t) ∈ R6 is defined as Ω = (ωb eR )T 01×3 . A. Velocity Filter Design To account for the lack of spacecraft translation and angular velocity measurements viz., V and ωb , or equivalently the velocity tracking errors viz., eV and ω, a filtered velocity error signal ef (t) ∈ R6 is produced using a filter. The filter is constructed as shown below ef = −kr0 + p,
(33)
where k > 0 is a positive, constant filter gain, p(t) ∈ R6 is a pseudo-velocity tracking error generated using p˙ = − (k + 1) p + k 2 r0 + Γr0 + ∆, p(0) = kr0 (0), (34) where Γ(t) ∈ R6×6 and ∆(t) ∈ R6 are defined as k1 (k P ef2 + eε diag{k I , I } and ∆ Γ 0 3 (1−eT e )2 3 = = ε ε T × −Cbd ωd eR )T 01×3 , respectively, k0 > 0 is a constant, k1 > 1 is a constant, and ef2 (t) is obtained by T decomposing ef as ef = eTf1 eTf2 with ef1 (t), ef2 (t) ∈ R3 . To assist in the development of the filtered velocity error signal ef dynamics, we introduce an auxiliary tracking error variable η(t) ∈ R6 as follows η
=
v0 + ef + r0 .
(35)
Note that using (35) in (32) produces r˙0 = η − ef − r0 − Ω. (36) T T T Next, we decompose η as η = η1 η2 , where η1 (t), η2 (t), ∈ R3 . Using this decomposition, (35) yields η1 = eV + ef1 + eR ,
η2 = e˙ ε + ef2 + eε .
(37)
In addition, solving for ωb in (20) and substituting for ω from (30) in the resulting equation, we obtain ωb = P (η2 − ef2 − eε ) + Cbd ωd ,
(38)
where (37) has been used. Next, we substitute (38) into the definition of Ω and rearrange terms to decompose Ω as Ω = Ω 1 + Ω2 , where ((−P
Ω1 (t), Ω2 (t) ∈ R6 ,
× T T Ω1 = [((P η2 ) eR ) 01×3 ] T d × T ef2 + eε + Cb ωd ) eR ) 01×3 .
and
(39) Ω2
=
To obtain the closed-loop dynamics of ef , we take the time derivative of (33), which yields (40) e˙ f = −kη − ef + Γr0 + kΩ1 , where (33), (34), (36), and ∆ = −kΩ2 have been used. B. Open-Loop Auxiliary Tracking Error Dynamics We begin by differentiating η of (35) with respect to time and substituting the time derivative of (30) and (31) to produce e˙ V η˙ = (41) + e˙ f + v0 − Ω, ˙ T ω + T ω˙
where (32) has been used. Next, we multiply M ∈ R6×6 T defined as M = diag{mI3 , P JP } on both sides of (41) to yield me˙ V M η˙ = T + M (e˙ f + v0 − Ω) . (42) T ˙ P JP T ω + P JP T ω˙
Using (24) and (29), P T JP T ω˙ in (42) is expressed as P T JP T ω˙ = P T (Jω)× ω +P T (τun + τkn + τ ). Next, we substitute for ω from (30) into P T JP T˙ ω + P T JP T ω˙ to obtain P TJP T˙ ω+P TJP T ω˙ = (P TJP T˙ P+P T(JP e˙ ε )× P )e˙ ε +P T (τun + τkn + τ ) .
(43)
To simplify notation, we define an inertia-like matrix T J (t) ∈ R3×3 as J (eε , eζ ) = P JP and a coriolis-like 3×3 ˙ matrix C (t) ∈ R as C (eε , eζ , e˙ ε , e˙ ζ ) =J TP + × T P (JP e˙ ε ) P . Using these definitions, (43) is given by P T JP T˙ ω + P T JP T ω˙ = C η2 − C ef + eε 2
+P T (τun + τkn + τ ) , (44) where (37) has been used. Next, to simplify e˙ f + v0 − Ω term in (42), we use (39), (40), and v0 from (35) to produce ¯ − 2ef + Γr ¯ 1 − Ω2 , (45) ¯ 0 + kΩ e˙ f + v0 − Ω=−kη ¯ where k¯ = k − 1 and Γ = Γ − I6 . Finally, using (10), (44), and (45), the open-loop dynamics η of (42) yields ¯ η+N +M (Γr ¯ Ω1 +u,(46) ¯ 0−2ef −Ω2 )+χ+kM M η=Λ− ˙ kM where Λ(t), N (t), χ(t), u(t) ∈ R6 are T as Λ η2 )T , N = 01×3 = defined (C mµ mµ 3µ T T RT − R
d
3
Rd +
T JR I3 − 5R R2
R3
R +
T
2R4
R
tr (J) I3 + 2J
× T (P τkn ) , χ = [(−mωb eV ) T T f (P T τ )T . eε ))T ]T , and u = T
T
T
(P τun
−C (ef2 + Remark 4.1: The inertia- and coriolis-like matrices of J and C satisfy the skew-symmetric property of T 1 ˙ z = 0, ∀z ∈ R3 . See [4] for details. z 2J + C C. Stability Analysis To facilitate the following stability analysis, we introduce several variables. We error T define TanTauxiliary T variable y(t) ∈ R6 as y = y1T y2T , = η1 (P η2 ) where y1 (t), y2 (t) ∈ R3 . Next, we define a combined er√ T T k1 eε T T T √ e e y , ror variable x(t) ∈ R18 as x f R = T 1−eε eε
ˆ ˆ ∈ R6×6 as M and a constant matrix M = diag{mI3 , J}. In addition, we define χ1 (t) ∈ R6 and χ2 (t) ∈ R3 as T (P η2 )× T I3 03×3 T ¯ ke + kη M . χ1 χ, χ T −1 2= f = 0 0 (P ) 3×3
3×3
(47) Finally, we define positive constants, λ1 , λ2 , keR , ˆ ky , and kmax as λ1 1 min{k0 , 1, m, λmin {J}}, k, =2 1 keR 1, λ2 = 2 max{k0 , 1, m, λmax {J}}, = k0 − ˆ ¯ ˆ ˆ k = kλmin {M }, ky = k − 1, and kmax = max keR , ky , respectively, where λmin {X} and λmax {X} represent the minimum and maximum eigenvalue, respectively, of a matrix X. Theorem 4.1: The output feedback control law u(t) given by k0 e R ¯ 0 − 2ef − Ω2 ) − k1 eε u = kef − N − M (Γr ,(48) (1−eTε eε )2
ensures semi-global asymptotic convergence of the spacecraft translational position and velocity tracking errors and the spacecraft attitude position and velocity tracking errors, delineated by lim eR (t), eV (t), eε (t), ω(t) = 0, if the initial condition t→∞ of eζ is selected such that eζ (0) = 0 and k, k0
λ2 are selected such that kmax > ρ21 λ1 x(0) λ 2 +ρ22 where ρ1 (·) and ρ2 (·) are λ1 x(0) , nondecreasing functions. Proof. We begin by substituting (48) into (46) to obtain the closed-loop dynamics for η k0 e R ¯ ¯ k e M η˙ = Λ − kM η + χ + kM Ω1 + kef − .(49) 1 ε
(1−eTε eε )2
Next, we define a positive-definite, candidate Lyapunov function as 1 1 T 1 k1 eTε eε 1 T ˆ y. (50) V + yT M = k0 eR eR + ef ef + 2 2 2 1 − eTε eε 2 Applying Rayleigh-Ritz’s theorem on (50) results in λ1 x2 ≤ V ≤ λ2 x2 .
(51)
Next, we compute the time derivative of (50) to produce V˙ = k0 eTR e˙ R + eTf e˙ f +
k1 eTε e˙ ε (1 −
2 eTε eε )
ˆ y. ˙ + yT M
(52)
ˆ y˙ on the right hand side of (52) is Note that y T M simplified as follows ˆ y˙ = η T M η˙ + 1 η T J˙ η2 , yT M (53) 2 2 where we used the time derivative of J defined earlier. Evaluating (52) along the trajectories of (36), (40), (49), and (53), we get V˙ = −k0 eTR eR − eTf ef −
k1 eTε eε 2
(1 − eTε eε )
¯ TM ˆy − ky
+y T χ1 + eTR χ2 , (54) where the definitions of (47) and the skew-symmetry property in Remark 4.1 have been used. Based on (47), it can be shown that χ1 and χ2 satisfy χ1 ≤
ρ1 (x)x and χ2 ≤ ρ2 (x)x, respectively, using which, (54) can be upper bounded as follows k1 eε 2 2 ˆ V˙ ≤ −k0 eR 2 − ef 2− 2 − ky (1 − eTε eε ) +ρ1 (x)xy + ρ2 (x)xeR , (55) where the definition of kˆ has been used. Then, using the definition of keR and ky yields the following expression for (55) V˙ ≤ −x2 − ky y2 + ρ1 (x)xy
V. Conclusion In this paper, we addressed an output feedback tracking control problem for a spacecraft with coupled translation and attitude motion when only translational position and attitude orientation measurements are available. A Lyapunov based tracking controller was designed with guaranteed semi-global, asymptotic stability for the position and velocity tracking errors. This control design methodology required only translation and attitude position measurements while estimating translation and angular velocity errors through a high pass filter.
D1
−keR eR 2 + ρ2 (x)xeR , D2
where the definition of x has been used. Bounding D1 and completion of squares produces V˙ ≤ D2 2in (56) by ρ1 (x) ρ22 (x) 2 x . Note that if kmax is chosen − 1− 4ky − 4ke R
such that kmax ≥ semidefinite, i.e.,
ρ21 (x)+ρ22 (x) , 4
then V˙ is negative
V˙ ≤ −βx2 , (57) where β is some positive constant. Utilizing (51) yields a sufficient condition for (57) as follows
V (t) V (t) 2 1 2 2 ˙ V ≤−βx , if kmax > 4 ρ1 + ρ2 . (58) λ1 λ1 Since V (x(t)) ≥ 0 and V˙ (x(t)) ≤ 0, we conclude that 0 ≤ V (x(t)) ≤ V (x(0)) < ∞. (59) Using (51) and (59) yields a sufficient condition for λ2 1 2 2 ˙ (58) given by V ≤ −βx , kmax > 4 ρ1 λ1 x(0) λ2 + 14 ρ22 λ1 x(0) . From (59), we know that V ∈ L∞ , thus eR , ef , √ eε T , η ∈ L∞ . Since √ eε T ∈ L∞ , we can 1−eε eε
1−eε eε
conclude that eε (t) < 1 for all time. Next, use of the unit norm constraint eTε eε + e2ζ = 1 with eε (t) < 1, ∀t ≥ 0, reveals that eζ (t) = 0, ∀t ≥ 0. In addition, it follows from (29) that det(T ) = eζ , using which, we conclude that T is invertible for all time. Since eR , ef , eε , η ∈ L∞ , it follows from (35) that v0 ∈ L∞ ; hence, due to the bound of Rd , R˙ d , εd , and ωd , we can use (4), (30), (38), (40), and (49) to conclude that R, eV , V, ωb , ω, d ( √ eε T ), e˙ f , η˙ ∈ L∞ . Similar signal dt 1−eε eε chasing arguments can now be employed to show that all other signals in the closed-loop system remain bounded. Using (57) and (59), it can be easily shown that eR , √ eε T , ef , η ∈ L2 . Since we have already shown that 1−eε eε eR , √ eε T , ef , η ∈ L∞ , we can utilize Barbalat’s 1−eε eε
Lemma to conclude that lim eR (t), √ t→∞
References
(56)
eε (t)
1−eε (t)T eε (t)
[1] J. Ahmed, V. T. Coppola, and D. S. Bernstein, “Adaptive asymptotic tracking of spacecraft attitude motion with inertia matrix identification,” AIAA J. GCD, vol. 21, pp. 684–691, 1998. [2] D. M. Boskovi´c and M. Kristi´c, “Global attitude/position regulation for underwater vehicles,” in Proc. ICCA, pp. 1768–1773, 1999. [3] V. A. Chobotov, Orbital Mechanics. Washington, DC: AIAA, 1996. [4] B. T. Costic, D. M. Dawson, M. S. de Queiroz, and V. Kapila, “A quaternion-based adaptive attitude tracking controller without velocity measurements,” AIAA J. GCD, vol. 24, pp. 1214–1222, 2001. [5] D. M. Dawson, J. Hu, and T. C. Burg, Nonlinear Control of Electric Machinery. New York, NY: Marcel Dekker, 1998. [6] P. C. Hughes, Spacecraft Attitude Control. New York: Wiley, 1986. [7] F. Lizarralde and J. T. Wen, “Attitude control without angular velocity measurement: A passivity approach,” IEEE TAC, vol. 41, pp. 468–472, 1996. [8] H. Pan and V. Kapila, “Adaptive nonlinear control for spacecraft with coupled translational and attitude dynamics,” in Proc. ASME Int. Mech. Eng. Congress and Exposition, Paper No. DSC-24580, 2001. [9] D. T. Stansbery and J. R. Cloutier, “Position and attitude control of a spacecraft using the state-dependent Riccati equation technique,” in Proc. ACC, pp. 1867–1871, 2000. [10] B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation. New York, NY: Wiley-Interscience, 1992. [11] F. Terui, “Position and attitude control of a satellite by sliding mode control,” in Proc. ACC, pp. 217–221, 1998. [12] H. Wong, M. S. de Queiroz, and V. Kapila, “Adaptive tracking control using synthesized velocity from attitude measurements,” in Proc. ACC, pp. 1572–1576, 2000. [13] Q. Yan, G. Yang, V. Kapila, and M. S. de Queiroz, “Nonlinear dynamics, trajectory generation, and adaptive control of multiple spacecraft in periodic relative orbits,” in Proc. AAS Guid. and Contr. Conf., Vol. 104, pp. 159–174, 2000.
Fi
f
R
Fb Spacecraft
,
ef (t), η(t) = 0, which is then used to show that lim eε (t) = 0. Finally, using (30), (31), and (35) it t→∞ is immediate that lim eV , ω = 0. Thus, the result of t→∞ Theorem 4.1 follows.
Fig. 1.
Schematic representation of the spacecraft system
